For the Waffles:
In large bowl, whisk together flour, baking powder, baking soda and salt. In medium bowl, whisk together buttermilk, eggs and melted butter. Add wet ingredients to dry and stir until lump free and well-incorporated. Set aside for 5 minutes while your waffle iron heats up. Add about 1/3 cup batter to hot waffle iron and cook according to machine directions until golden brown and crispy. Transfer cooked waffle to wire rack and continue cooking rest of batter.
Heat stovetop grill pan over medium-high heat. Season both sides of chicken breasts with salt, pepper and cayenne. Butter grill pan and cook chicken in batches for about 2 to 4 minutes per side or until done all the way through. Transfer to cutting board to rest for about 10 minutes. Thinly slice and set aside.
For the Grilled Cheese Sandwiches:
Spread each waffle with some pesto, then lay 3 slices of mozzarella on top of each waffle. Place sliced chicken, red peppers and basil on top of three waffles, then sandwich together with other three waffles that have just pesto and cheese on them.
Heat griddle or nonstick skillet over medium-low heat. Melt 1 tablespoon butter and place one waffle sandwich in butter. Cook until golden brown and crispy on first side, about 2 to 3 minutes. Flip over and cook for another 2 minutes. Once cheese has melted, transfer to wire rack and keep warm in 250°F oven. Add another tablespoon of butter to griddle and continue cooking other waffle sandwiches on both sides. Cut into wedges and serve immediately, 2 triangles per serving.